Minimal Faithful Quasi-Permutation Representation Degree of p-Groups with Cyclic Center

TL;DR

This paper investigates the minimal degrees of faithful permutation and quasi-permutation representations for certain classes of finite non-abelian p-groups with cyclic centers, extending known results and providing new computations.

Contribution

It generalizes previous results for nilpotency class 2 p-groups to normally monomial p-groups with cyclic centers and computes minimal degrees for some metabelian p-groups.

Findings

Generalization of Behravesh's result to normally monomial p-groups

Explicit minimal degree computations for specific metabelian p-groups

Enhanced understanding of representation degrees in p-groups with cyclic centers

Abstract

For a finite group G, we denote by $\mu(G)$, and c(G), the minimal degree of faithful permutation representation of G, and the minimal degree of faithful representation of G by quasi-permutation matrices over the complex field C, respectively. In this article, we study $\mu(G)$, and c(G) for various classes of finite non-abelian p-groups with cyclic center. We prove a result for normally monomial p-groups with cyclic center which generalizes a result of Behravesh for finite p-groups of nilpotency class 2 with cyclic center [5, Theorem 4.12]. We also compute minimal degrees for some classes of metabelian p-groups.